Skip to main content

Full text of "The Kinematics of Machines."

See other formats

1895.] The Kinematics of Machines. lr>7 

The distances between the junctions were BM = 2'57 cm. and MT = 
2*6 cm. Hence by the formula of § 2, 

&(M,JB) = 73-1 -4-2 -6 = 28-1 __ 
fc(T, M) ~~ 79*0-7- 2*57 307 

Aberdeen granite : 

t>(T) = 81°-1. 

v(M) = 145°"6. 

i?(B) = 214°-6. 

The distances between the junctions were BM = 1*9 cm. and 
MT = 2-0 cm. 

K M B) — 64-5-7-2-0 _ 32*2 

A;(TM) 69-0-7-1-9 36*3 


§ 13. Thus we see, that for slate, with lines of flux parallel to 
cleavage planes, the mean conductivity in the range from 123° C. to 
202° C. is 91 per cent, of the mean conductivity in the range from 
50° C. to 123° C, and for granite, the mean conductivity in the range 
from 145° C. to 214° C. is 88 per cent, of the mean conductivity in 
the range from 81° C. to 145° C. The general plan of apparatus, 
described above, which we have used only for comparing the con- 
ductivities at different temperatures, will, we believe, be found 
readily applicable to the determination of conductivities in absolute 

II. " The Kinematics of Machines." By T. A. Hkarson, 
M.Inst.C.E., Professor of Mechanism and Hydraulic Engi- 
neering, Royal Indian Engineering College, Coopers Hill. 
Communicated by Professor Cotterill, F.R.S. Received 
March 19, 1895. 


In this paper the author regards a machine as an embodiment of 
a movement. The method of construction and the proportions of the 
parts are not taken into consideration, except so far as may be neces- 
sary to explain the conditions requisite for the kinds of motions with 
which they are supposed to be endowed. All other considerations 
relating to form and proportion are omitted, as belonging to the sub- 
ject of machine design. Neither does the author take account of the 
forces which actuate and oppose the movement of the machine, such 
matters belonging to the subject Dynamics of Machines. 

The object of the paper is to analyse the movements only, and to 

168 Prof. T. A. Hearson. [May 30, 

show the likeness and the differences between machines in similarities 
in the movements or the contrary. 

It is claimed by the author that in those movements the principal 
feature of a machine resides, distinguishing it from other engineering 

It is shown that all movements, however complex, are derived 
from the association together of some of a comparatively limited 
number of kinds of more or less simple motions, which take place 
between consecutive directly connected pieces. 

Certain geometrical laws are enunciated, from which are derived 
the conditions necessary for the association of those motions together 
in one machine. It is shown that those laws preclude the existence 
of certain combinations of motions, and it is suggested that one may 
be enabled by this analysis to enumerate an exhaustive list of the 
possible combinations which must include all existing machines, and 
suggest the design of others not in existence. Moreover, by attaching 
to each kind of motion a suggestive symbol, a method of expressing 
the constitution of a machine movement by a simple formula is 
proposed, whereby similarities and differences between machines may 
be exhibited at a glance. 

The author commences by considering a very simple mechanism, 
consisting of four bars united in one continuous linkage by four pins 
which have parallel axes. By imagining the length of the links to 
undergo variation from zero to infinity, it is shown that this simple 
mechanism is representee of all the simple plane mechanisms, and, 
by imagining other variations to occur, this same mechanism is 
shown to be representive of still further classes of mechanisms, 
in which the parts do not move in or parallel to one plane. In this 
simple mechanism the relative motions of consecutive pieces are 
either turning, when one piece revolves completely around relatively 
to the other, the representative symbol being the letter O, or swing- 
ing when one piece turns through a limited angle relatively to the 
adjoining one, represented by the letter U. 

The first law enunciated, which governs the association of the O 
and U motions, is founded on the geometrical fact that the sum of 
the three angles of a plane triangle is constant, and the sum of the 
four angles of the quadrilateral therefore also constant. After a 
complete revolution the angle between the bars is considered to have 
been increased or diminished by 'lir. With this extension of the pro- 
position the constancy of the sum of the angles is unimpaired. 

From this it is seen to be impossible for only one motion to be turn- 
ing and the other three swinging, otherwise the sum of the four 
angles would increase or decrease by 2w each revolution. 

The second law, which governs the association of the motions, has 
to do with the proportions between the length of the links necessary 

1895.] The Kinematics of Machines. 169 

to permit of complete turning. This is founded on the fact that 
one side of a triangle cannot be greater than the sum of the other 
two. From these two laws together it is shown that it is impossible 
to have two O's alternating with two U's. 

Next it is pointed out how the U motion may be provided for by 
constructing a circular slotway in one piece, and shaping the other 
piece to fit the slotway, so that by imagining the radius of curvature 
of the slotway to be indefinitely increased a relative movement of 
reciprocating sliding motion, represented by the symbolical letter I, 
will be substituted for the swinging motion U. A slide being con- 
ceived to be a swing through a zero angle about an infinitely distant 
centre, the previously mentioned laws will apply to associations con- 
taining I motions, and it will follow that a- combination of three 
slides and one swing is precluded by the first law. 

If four slides are associated, in which all four of the links of 
the original mechanism are to be conceived to be infinitely long, 
an indeterminate motion will result comparable to the motion which 
would be possible if five bars were joined by pairs in a closed circuit. 

One of the slides may be suppressed, and a definite motion will 
result from three slides. 

If the foregoing analysis be compared with that due to Reuleaux, 
to which it bears a close resemblance, it will be seen that Reuleaux 
conceives that the elementary essential components of machines are 
the pairs of consecutive links which are in mutual contact, whereas 
it is here proposed that the relative motions of consecutive links 
should be regarded as the essential elements or components of a 
machine movement. Whilst the pairs of surfaces of contact of con- 
secutive pieces should be formed to suit the kind of relative motion 
which those pieces are required to undergo, yet the forms of those 
surfaces do not themselves entirely govern the character of the 

Reuleaux assumes that what he calls a turning motion and the I 
motion are entirely governed by the forms of the surfaces of mutual 
contact, but shows that to ensure a more complex motion a restraint 
is required to be imposed by means external to the two links. Those 
additional means of constraint have to be included with that due to 
the forms of the surfaces of mutual contact in the conception of a 
complete pair, and often the whole mechanism is required to complete 
one pair contained in it. 

Reuleaux does not attempt to discriminate between a turning and 
a swinging pair ; the same pair of surfaces of mutual contact is 
suitable for both ; the difference consists of a difference only in the 
rest of the mechanism, yet the difference in the two motions is most 
apparent, and is very important, both kinematically and also from 
the practical engineer's point of view. 

170 Prof. T. A. Hearson. [May 30, 

No advantage is" derived from analysing a machine into parts such 
as pairs if it requires the whole machine to complete one of the 

The enunciation and the explanation of the influence of the first 
law previously mentioned, of the constancy of the sum of the four 
angles of a quadrilateral in governing the association of the OU and 
I motions in one mechanism, is one of the important original features 
of this paper. 

The influence of the second law, viz., that the two sides of a 
triangle are together not less than the third, in limiting the associa- 
tion of the 0, U, I motions, is now also for the first time pointed out, 
though Heuleaux and others have, without formally enunciating the 
law, made use of the fact to determine the proportions necessary for 
certain suggested movements. 

By the application of these governing laws one is able to write 
an exhaustive list of all the possible combinations in one simple 
mechanism of the three simple O, U, I motions, and to explain why 
other combinations are precluded. 

Fourteen distinct combinations are possible, and only fourteen. 
They are exhibited by the following formulae, in which a large O 
associated with a small o signifies that in one case adjacent links 
turn relatively to one another so as to continuously increase the 
angle between them, and in the other to continuously diminish the 
angle. The double @ signifies that two complete revolutions accom- 
pany one complete to and fro swing or slide. 


Group. Q 


V Group. r 1 C <J W Q 

Following Reuleaux, the author applies the principle of the " in- 
version of the kinematic chain," considering' it to be a continuous 
sequence of links in a closed circuit containing a sequence of 
elementary motions. In explaining what is meant by inversion, \t is 
pointed out that relatively to an observer or user of a machine one 
piece is fixed. This is called the frame of the machine. Bach one of 
the four links may in turn be made the fixed or frame link, and 

1895*] The Kinematics of Machines. 171 

although, the relative motion of the four links will in all cases remain 
unaltered, the absolute movement, or movement relatively to the 
user of the machine, will in general be different for each fixing, and 
constitute a new machine movement. Changing the fixed or frame 
link is called the " inversion of the chain. " 

The author makes use of the term " primary pieces/' originally 
suggested by Rankine for those links which are in sequence with 
and directly connected to the frame link, and shows that if, after 
inversion, the new primary pieces have the same kind of motions as 
the previous primary pieces had, the consequent machine movement 
is not a new one, but a repetition of a previous one. 

From the mechanisms C ) and f~ J only can four different 

machine movements be obtained by inversion. From the others 3, 2, 
or only 1 can be derived. 

They are distinguished from one another in the formula by using 
a thick line for the frame link. Thus 

O signifies a machine movement like that employed in the 
crank and connecting-rod engine. 

fr~J is exemplified in the oscillating engine much used in paddle- 
^~~± wheel steamers. 

<£J> is f-nd in StannaVs pendulum pump, and 

(zTZ) is the movement adopted by Rigg in the design of his high 
Xh ~ :r speed engine. The intimacy of the relation of this engine 
to the preceding ones is here for the first time indicated. 

In all, thirty-two and only thirty-two distinct machine movements 
can be derived from the fourteen previously enumerated mechanisms 
by inversion. 

It is shown that Reuleaux's principle of inversion can be applied 
with more advantage and consistency if a machine movement is 
analysed into its component motions than if a machine is analysed 
into its component pairs, and the notation lends itself to a very clear 
exhibition of the effect of inversion. 

The author next discusses the relation of cams and spur-wheel 
mechanisms to the foregoing kinematic chains, showing that they 
are the result of the suppression of one of the previous four links and 
the amalgamation of the two adjoining simple motions into one more 
complex. A comparison is also made with belt gearing and expres- 
sive formulae suggested. 

The author then passes to the consideration of machines the parts 
of which do not move parallel to one plane. 

172 Prof. T. A. Hearson. [May 30, 

Reuleaux was the first to show that if the links of the previously 
mentioned kinematic chains be bent to the form of great circles of a 
sphere the axes of the connecting pins will be radial, and the pre- 
viously mentioned machine movements will be possible under the 
modified circumstances. 

In spherical motion the counterpart of what is a slide in plane 
motion could be obtained by a swinging motion about a pole of 
which the bent link is the equator. The motion is to be conceived 
as due to the use of two bent links, the length of one of which is a 
quadrant of a great circle of the sphere. 

In these so-called spherical mechanisms, Law I has to be modified 
as follows : — 

The sum of the four angles of the spherical quadrilateral varies, 
having a value of S7r for a maximum and 2tt for a minimum. 

This and Law II, which is the same as before, will preclude the 
same combinations in spherical mechanisms which were precluded in 
plane mechanisms. 

Law I explains at once why in Hooke's joint, which is the spherical 
counterpart of Oldham's coupling, the angular velocity-ratio of the 
connected shafts is not constant, whereas in Oldham's coupling it is. 

The author points oat that the kinematic chain containing three 
slides cannot be adapted to give a movement on a sphere. The 
virtual construction would consist of a spherical triangle between 
the links of which no relative motion is possible, and there is not 
room on the sphere for a movement at each joint of a bent quadri- 
lateral, the length of each side of which is equal to a quadrant. But 
a three-slide mechanism can be adapted to give motion on the surface 
of a cylinder, and it is the only one of the fourteen kinematic chains 
which can be so adapted, and examples of it are found in the various 
helical motions so largely used. (The letter V is used to represent 
helical motions.) This method of showing the relation between 
screw motions and plane motions is a novel feature of the paper. 

The remaining mechanisms consist of those in which the axes of 
the turning and swinging motions neither meet nor are parallel. 
They include the motion which occurs at a ball-and-socket joint 
represented by @. The method of classification according to the 
proposed scheme is summarised as follows : — 

All simple machine movements may be ranged in four divisions, 
viz. : — 

1. Consisting of plane mechanisms, in which the pieces move in or 
parallel to the surface of a plane. 

2. Spherical mechanisms, in which the pieces move in or parallel 
to the surface of a sphere. 

3. Cylindrical mechanisms, in which the pieces move in or parallel 
to the surface of a cylinder. 

1895*] The Kinematics of Machines. 173 

4. The remainder, to which the name conoidal mechanisms is given, 
in which, the axes of the swinging and turning motions neither meet 
nor are parallel. 

The mechanisms in each of these divisions are classed in two sub- 

Sub-division S, with surf ace contact of consecutive links. 
Sub-division P, with point contact of consecutive links. 

The mechanisms of sub-division S of divisions 1 and 2, 1^ and 2 S 
will consist of those in which U I motions only are used. 

Those of 3 S will include the helical or V motion, and 

Those of 4< s will include the motion @ requiring the use of a ball- 
and-socket joint. 

To the pairs of links which have the relative motions O, U, I, V, 
Reuleaux has given the name lower pairs. Reuleaux claimed two 
characteristics for lower pairs, viz. : — • 

1. Definiteness of motion derived from the surfaces of mutual con- 
tact themselves. 

2. The possibility of distributing the contact over an area which 
may be extended as much as desired. 

If it is desired to differentiate between, the and U motions, 
Beuleaux's turning pair cannot possess the first characteristic. 

The second characteristic is of considerable value in relation to the 
liability to abrasion and wear, but the advantage of greater immunity 
against wear has to be purchased at the cost of a more complicated 
construction and a more restricted character of movement. 

As examples — 

The mechanism, consisting of a pair of spur wheels turning in 
bearings which are at a fixed distance apart will belong to 1^. 

A pair of bevel wheels will belong to 2 P . 

The so-called cylindrical cam motion will belong to 3^, and the 
worm-and-worm wheel mechanism to 4^. 

The mechanisms in each of the eight sub-divisions are still further 
sub-divided into combinations. The combinations of l s , 2 S , and 3^, 
are exhaustively enumerated, and it is suggested that an extension of 
the methods of applying the geometrical laws would lead to the 
preparation of an exhaustive list of the possible combinations in the 
other sub-divisions. The combinations are still further sub-divided 
into inversions according to Reuleaux's principle of the inversion of 
a machine. 

Further than this there will be varieties of any inversion differing 
in the details of the construction and uses of the machine move- 

Lastly, the author proceeds to snow how the foregoing considera- 
tions assist in the analysis of compound mechanisms. It is assumed 

1 74 Mr. W. E. Wilson. Effect of Pressure on the [May 30, 

that practically all compound mechanisms contain a continuous 
mechanism A, of not more than four links, from which definiteness of 
relative motion of all the other links is derived. Any two links 
of A in their exact length, or longer or shorter, may be adapted to 
form with two new links a second mechanism B, and any two of A or 
B, or one of A and one of B, may be adapted to form with two still 
further added links a third mechanism 0, and so on. In this way a 
definiteness of relative motion of many links in a compound 
mechanism is derived. The notation lends itself to a clear exhibition 
of the manner in which two or more simple mechanisms are associated 
together, and the compound mechanism built up. 

III. " On the Effect of Pressure of the Surrounding Gas on the 
Temperature of the Crater of an Electric Arc Light. 
Preliminary Notes of Observations made at Daramona, 
Streete, co. Westmeath." By W. E. Wilson. Commuin- 
cated by Professor Fitzgerald, F.R.S. Received April 
25, 1895. 

Of lafce years it has often been assumed that the temperature of 
the crater forming the positive pole of the electric arc is that of the 
boiling of carbon. The most modern determinations give this point 
as about 3300°— 3500° 0. 

Solar physicists have thought that the photosphere of the sun 
consists of a layer of clouds formed of particles of solid carbon. As 
the temperature of these clouds is certainly not below 8000° C, it 
seems very difficult to explain how carbon can be boiling in the arc 
at 3500° and yet remain in the solid form in the sun at 8000°. Pres- 
sure in the solar atmosphere seemed to be the most likely cause of 
this, and yet, from other physical reasons, this seemed not probable. 

In order to investigate whether increased pressure in the gas sur- 
rounding an electric arc would raise the temperature of the crater, I 
had an apparatus made by the Cambridge Instrument Company. It 
consists of a strong cast-iron box, which was tested by hydraulic 
pressure to 2000 lbs. per square inch. In the following plan, A is 
the box, B and C are the two carbon poles enclosed in steel tubes. 
The negative carbon was kept in position against a copper ring by a 
spiral spring behind it. The positive carbon was hand fed by a 
friction roller, which was moved by a handle F outside the box. A 
steel tube H was screwed into the box at such an angle that, by 
looking down it, we could see well into the crater of the positive pole. 
The end of this tube is closed by a glass lens, which formed an image 
of the crater at a distance of 80 cm.